Fourier Optics - Impulse Response of Free Space from Fresnel Transfer Function

Question

I am currently reading the chapter "Fourier Optics" in the book "Fundamentals of Photonics" by Saleh and Teich. However I am not able to follow one specific mathematical derivation.

On page 111 the transfer function of free space is derived $$ H(\nu_x, \nu_y) = \text{exp}(-j 2 \pi d \sqrt{\lambda^{-2} - \nu_x^2 - \nu_y^2}).$$

$d$ is the distance the light travels from the input plane to the output plane. $\lambda$ is the wavelength and $\nu_x$ and $\nu_y$ are the spatial frequency components.

After that this formula is being simplified by using the fresnel approximation, for which it is assumed, that the frequency components $\nu_x$ and $\nu_y$ in the input wave are much smaller than the system bandwidth $\lambda^{-1}$. The resulting approximated transfer function is $$ H_{\text{Fresnel}}(\nu_x, \nu_y) = \text{exp}(j \pi \lambda d (\nu_x^2 + \nu_y^2)) \cdot \text{exp}(-j k d).$$

This still makes sense to me, everything is fine so far. However after that they derive the impulse response of the system by applying the inverse fourier transform to the transfer function $H_{\text{Fresnel}}$. The resulting function is $$h(x,y) \approx \dfrac{j}{\lambda d} \cdot \text{exp}(-j k d) \cdot \text{exp}(-j k \dfrac{x^2+y^2}{2 d}).$$

And honestly, I have absolutely no idea how they come to that expression. The inverse fourier is $$h(x, y) \approx \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} H_{\text{Fresnel}}(\nu_x, \nu_y) \cdot \text{exp}(-j 2 \pi (\nu_x x + \nu_y y)) d\nu_x d\nu_y.$$

Small annotation: Out of some reason they flipped the signs in the fourier transform in contrast to the standard notation.

So the core question is: How did they solve this integral? There is a correspondence table at the end of the book, but I have no clue how this should help. Kind regards

Answer 1

I think I was able to solve the problem by applying the same method as mentioned here. However my solution still differs by a constant factor from the solution in the book, so maybe it is still not completely right.

If you look at $h(x, y)$ one can see easily that it can be separated as $$h(x, y) = K \cdot f(x) \cdot f(y).$$ with $f(x) = e^{\dfrac{-j k x^2}{2 d}} = e^{j a x^2}, a = \dfrac{-k}{2 d}$ and $K = e^{-j k d}$.

So if we know how to fourier transform $f(x)$, the problem is more or less solved.

If we differentiate $f(x)$, we get the following equation $$\dfrac{d f(x)}{dx} = f(x) \cdot 2 j a x.$$

Lets fourier transform it with the known correspondences $$j \omega F(\omega) = \dfrac{d F(\omega)}{d \omega} 2ja.$$

This gives us $$\dfrac{d F(\omega)}{d \omega} = F(\omega) \cdot \dfrac{\omega}{2 a j}.$$

We can see, that $$F(\omega) = \text{exp}(\dfrac{-j \omega^2}{4a})$$ is a solution of the differential equation.

No we can resubstitute $a$ and $k = \dfrac{2 \pi}{\lambda}$ $$F(\omega) = \text{exp}(\dfrac{j \omega^2 d \lambda}{4 \pi})$$ and with $\omega = 2\pi \nu$ we get $$F(\nu) = \text{exp}(j \pi \lambda d \nu^2).$$

Resubstituting everything gives us $$H(\nu_x, \nu_y) = \dfrac{j}{\lambda d} \text{exp}(-j k d) \text{exp}(j \lambda d \pi (\nu_x^2+\nu_y^2)).$$

Out of some reason the factor $\dfrac{j}{\lambda d}$ is still wrong, but that's the best answer i could derive.

Kind Regards

Answer 2

Remeber that for $a>0$ the Fresnel integral is $$ \int_{-\infty}^{\infty} e^{iax^2} = e^{i\pi/4} \sqrt{\frac \pi{a}}, $$
because of the need to push the contour off the real axis with $x= e^{i\pi/4}t$. Your integral has the product of two Fresnel integrals $x$ times $y$ and so you have $$ \left[e^{i\pi/4} \sqrt{\frac \pi{a}}\right]^2= \frac{\pi i}{a}. $$